In this work, we investigate the convergence properties of the backward regularized Wasserstein proximal (BRWP) method for sampling a target distribution. The BRWP approach can be shown as a semi-implicit time discretization for a probability flow ODE with the score function whose density satisfies the Fokker-Planck equation of the overdamped Langevin dynamics. Specifically, the evolution of the score function is computed using a kernel formula derived from the regularized Wasserstein proximal operator. By applying the Laplace method to obtain the asymptotic expansion of this kernel formula, we establish guaranteed convergence in terms of the Kullback-Leibler divergence for the BRWP method towards a strongly log-concave target distribution. Our analysis also identifies the optimal and maximum step sizes for convergence. Furthermore, we demonstrate that the deterministic and semi-implicit BRWP scheme outperforms many classical Langevin Monte Carlo methods, such as the Unadjusted Langevin Algorithm (ULA), by offering faster convergence and reduced bias. Numerical experiments further validate the convergence analysis of the BRWP method.

翻译：本研究探讨了用于采样目标分布的后向正则化Wasserstein邻近算子（BRWP）方法的收敛性质。BRWP方法可视为具有得分函数的概率流常微分方程的半隐式时间离散化，其密度满足过阻尼朗之万动力学的福克-普朗克方程。具体而言，得分函数的演化通过从正则化Wasserstein邻近算子导出的核公式进行计算。通过应用拉普拉斯方法获得该核公式的渐近展开，我们建立了BRWP方法向强对数凹目标分布收敛的Kullback-Leibler散度保证。我们的分析还确定了收敛的最优和最大步长。此外，我们证明了确定性和半隐式的BRWP方案相较于未调整朗之万算法（ULA）等经典朗之万蒙特卡洛方法具有更快的收敛速度和更小的偏差。数值实验进一步验证了BRWP方法的收敛性分析。